#2032740

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,10,11)(7,9,8,12)(13,14,16,18)(15,17,19,20)(21,22,26,29)(23,27,24,30)(25,33,34,28)(31,36,32,35);(6,11)(9,12)(15,19)(17,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(5,7,10,8)(6,9,11,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,14)(15,17)(16,18)(19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,9,10,12)(6,7,11,8)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1939936);